On complexification and iteration of quasiregular polynomials which have algebraic degree two

Ewa Ligocka¹

¹ Institute of Mathematics Department of Mathematics, Computer Science and Mechanics Warsaw University Banacha 2 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 186 (2005) no. 3, pp. 269-285.

Résumé

We prove that each degree two quasiregular polynomial is conjugate to $Q(z)=z^{2}-(p+q)|z|^{2}+pq\overline{z}^{2}+c$, $|p|1$, $|q|1$. We also show that the complexification of $Q$ can be extended to a polynomial endomorphism of $\mathbb{C}\mathbb{P}^{2}$ which acts as a Blaschke product $\frac{z-p}{1-\overline{p}z}\cdot \frac{z-q}{1-\overline{q}z}$ on $\mathbb{C}\mathbb{P}^{2}\setminus\mathbb{C}^{2}$. Using this fact we study the dynamics of $Q$ under iteration.

Détail

DOI : 10.4064/fm186-3-5

Keywords: prove each degree quasiregular polynomial conjugate overline complexification extended polynomial endomorphism mathbb mathbb which acts blaschke product frac z p overline cdot frac z q overline mathbb mathbb setminus mathbb using study dynamics under iteration